AN IDENTITY BETWEEN THE m-SPOTTY ROSENBLOOM-TSFASMAN WEIGHT ENUMERATORS OVER FINITE COMMUTATIVE FROBENIUS RINGS

Abstract

Abstract. This paper is devoted to presenting a MacWilliams type iden-tity for m-spotty RT weight enumerators of byte error control codes overfinite commutative Frobenius rings, which can be used to determine theerror-detecting and error-correcting capabilities of a code. This providesthe relation between the m-spotty RT weight enumerator of the code andthat of the dual code. We conclude the paper by giving three illustrationsof the results. 1. IntroductionThe error control codes play an important role in improving reliability incommunications and computer memory system [5]. Recently, high-densityRAM chips with wide I/O data, called a byte, have been increasedly usedin computer memory systems. These chips are very likely to have multiplerandom bit errors when exposed to strong electromagnetic waves, radio-activeparticles or high-energy cosmic rays. To make these memory systems morereliable, spotty [21] and m-spotty [20] byte errors are introduced, which can beeffectively detected or corrected using byte error-control codes. To make clearthe error-detecting and error-correcting capabilities of a code, the research hasbeen done on some special types of polynomials, called weight enumerators.In general, the weightenumeratorofacode is apolynomialdescribingcertainproperties of the code, and an identity which relates the weight enumerator of acode with that of its dual code is called the MacWilliams type identity. For thepast few years, various weight enumerators with respect to m-spotty HammingWeight (Lee weight and RT weight) have been studied for various types ofcodes. Suzuki et al. [19] proved a MacWilliams type identity for binary byteerror-controlcodes. M. Ozenand V. Siap[8] andI. Siap[17] extended this result¨